lahef#

Functions

void clahef(
    const char*          uplo,
    const INT            n,
    const INT            nb,
          INT*           kb,
          c64*  restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          c64*  restrict W,
    const INT            ldw,
          INT*           info
);
void clahef(const char *uplo, const INT n, const INT nb, INT *kb, c64 *restrict A, const INT lda, INT *restrict ipiv, c64 *restrict W, const INT ldw, INT *info)#

CLAHEF computes a partial factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method.

The partial factorization has the form:

A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = ‘U’, or: ( 0 U22 ) ( 0 D ) ( U12**H U22**H )

A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) if UPLO = ‘L’ ( L21 I ) ( 0 A22 ) ( 0 I )

where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. Note that U**H denotes the conjugate transpose of U.

CLAHEF is an auxiliary routine called by CHETRF. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = ‘U’) or A22 (if UPLO = ‘L’).

Parameters

in
uplo

Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = ‘U’: Upper triangular = ‘L’: Lower triangular

in
n

The order of the matrix A. n >= 0.

in
nb

The maximum number of columns of the matrix A that should be factored. nb should be at least 2 to allow for 2-by-2 pivot blocks.

out
kb

The number of columns of A that were actually factored. kb is either nb-1 or nb, or n if n <= nb.

inout
A

Complex array, dimension (lda, n). On entry, the Hermitian matrix A. If uplo = ‘U’, the leading n-by-n upper triangular part contains the upper triangular part. If uplo = ‘L’, the leading n-by-n lower triangular part contains the lower triangular part. On exit, A contains details of the partial factorization.

in
lda

The leading dimension of the array A. lda >= max(1, n).

out
ipiv

Integer array, dimension (n). Details of the interchanges and the block structure of D. If uplo = ‘U’: Only the last kb elements of ipiv are set. If ipiv[k] >= 0, rows and columns k and ipiv[k] were interchanged, D(k,k) is a 1-by-1 diagonal block. If ipiv[k] < 0, rows and columns k-1 and -(ipiv[k]+1) were interchanged, D(k-1:k,k-1:k) is a 2-by-2 block, and ipiv[k-1] = ipiv[k]. If uplo = ‘L’: Only the first kb elements of ipiv are set. If ipiv[k] >= 0, rows and columns k and ipiv[k] were interchanged, D(k,k) is a 1-by-1 diagonal block. If ipiv[k] < 0, rows and columns k+1 and -(ipiv[k]+1) were interchanged, D(k:k+1,k:k+1) is a 2-by-2 block, and ipiv[k+1] = ipiv[k].

out
W

Complex array, dimension (ldw, nb). Workspace for storing updated columns during factorization.

in
ldw

The leading dimension of the array W. ldw >= max(1, n).

out
info

  • = 0: successful exit

  • > 0: if info = k+1, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular.

Functions

void zlahef(
    const char*          uplo,
    const INT            n,
    const INT            nb,
          INT*           kb,
          c128* restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          c128* restrict W,
    const INT            ldw,
          INT*           info
);
void zlahef(const char *uplo, const INT n, const INT nb, INT *kb, c128 *restrict A, const INT lda, INT *restrict ipiv, c128 *restrict W, const INT ldw, INT *info)#

ZLAHEF computes a partial factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method.

The partial factorization has the form:

A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = ‘U’, or: ( 0 U22 ) ( 0 D ) ( U12**H U22**H )

A = ( L11 0 ) ( D 0 ) ( L11**H L21**H ) if UPLO = ‘L’ ( L21 I ) ( 0 A22 ) ( 0 I )

where the order of D is at most NB. The actual order is returned in the argument KB, and is either NB or NB-1, or N if N <= NB. Note that U**H denotes the conjugate transpose of U.

ZLAHEF is an auxiliary routine called by ZHETRF. It uses blocked code (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = ‘U’) or A22 (if UPLO = ‘L’).

Parameters

in
uplo

Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = ‘U’: Upper triangular = ‘L’: Lower triangular

in
n

The order of the matrix A. n >= 0.

in
nb

The maximum number of columns of the matrix A that should be factored. nb should be at least 2 to allow for 2-by-2 pivot blocks.

out
kb

The number of columns of A that were actually factored. kb is either nb-1 or nb, or n if n <= nb.

inout
A

Complex array, dimension (lda, n). On entry, the Hermitian matrix A. If uplo = ‘U’, the leading n-by-n upper triangular part contains the upper triangular part. If uplo = ‘L’, the leading n-by-n lower triangular part contains the lower triangular part. On exit, A contains details of the partial factorization.

in
lda

The leading dimension of the array A. lda >= max(1, n).

out
ipiv

Integer array, dimension (n). Details of the interchanges and the block structure of D. If uplo = ‘U’: Only the last kb elements of ipiv are set. If ipiv[k] >= 0, rows and columns k and ipiv[k] were interchanged, D(k,k) is a 1-by-1 diagonal block. If ipiv[k] < 0, rows and columns k-1 and -(ipiv[k]+1) were interchanged, D(k-1:k,k-1:k) is a 2-by-2 block, and ipiv[k-1] = ipiv[k]. If uplo = ‘L’: Only the first kb elements of ipiv are set. If ipiv[k] >= 0, rows and columns k and ipiv[k] were interchanged, D(k,k) is a 1-by-1 diagonal block. If ipiv[k] < 0, rows and columns k+1 and -(ipiv[k]+1) were interchanged, D(k:k+1,k:k+1) is a 2-by-2 block, and ipiv[k+1] = ipiv[k].

out
W

Complex array, dimension (ldw, nb). Workspace for storing updated columns during factorization.

in
ldw

The leading dimension of the array W. ldw >= max(1, n).

out
info

  • = 0: successful exit

  • > 0: if info = k+1, D(k,k) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular.

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